bayroot: A Bayesian phylogenetic approach to dating HIV reservoir sequences

The latent HIV reservoir

• HIV provirus in the latent reservoir may come from an early or late stage of infection.
  • Adaptation to host-specific immune response, drug resistance.
• Is long-term persistence of the reservoir due to clonal expansion or ongoing replication?
  • Replication in anatomic compartments with low drug penetrance would replenish a “young” reservoir.

Problems with RTT dating

Ignores variation in number of mutations — there is one, and only one, estimated date. Unclear how to deal with divergent outgrowth sequences. Proviral sequences can be mapped to the future (right, red zone). Assumes the tree is known without error.

A simple Bayesian model

• The RTT regression model has three parameters:

  1. The date of the origin (\(x\)-intercept);
  2. The molecular clock (rate/slope, \(\mu\));
  3. The location of the root in the tree (determines divergence, \(y\))

• We will sample these parameters from the posterior distribution.

  • Continue to assume the unrooted tree is known without error (constrains positioning of root).

Poisson likelihood

• A strict clock assumes that mutations accumulate at a constant rate \(\mu\) over time, so variation is equal to the mean (Poisson).
  • Let \(Y_i\) be the number of mutations in the \(i^{\textrm{th}}\) sequence, given location of root.
  • Let \(\Delta t_i\) be the time elapsed between the \(i^{\textrm{th}}\) sample and the root.
• The log-likelihood for RNA set \(\{Y_i, \Delta t_i\}\) is:

\[\log L(Y_i, \Delta t_i) = \sum_i Y_i\log(\mu \Delta t_i) - \mu \Delta t_i - \log \Gamma(Y_i+1)\]

Metropolis-Hastings sampling

Sample run

Step 180 Step 19980

Sampling integration dates

• Using Bayes’ rule, the probability of integration time \(t_i\) for outgrowth sequence with divergence \(Y_i\) is: \[P(t_i|Y_i) = \frac{P(Y_i|t_i) P(t_i)}{P(Y_i)}\]

• We assume \(P(t_i) = \frac{1}{T-t_0}\), where \(t_0\) is the origin date and \(T\) is the maximum integration date, and letting \(s=t-t_0\):

\[ P(Y_i) = \frac{\int_0^{T-t_0} (\mu s)^{Y_i} \exp(-\mu s) \mathrm{d}s}{(T-t_0)\Gamma(Y_i+1)} = \frac{\gamma(Y_i+1, \mu(T-t_0))}{\mu(T-t_0)\Gamma(Y_i+1)} \]

where \(\gamma(s, x)\) is the lower incomplete gamma function.

Sampling integration dates (2)

• Letting \(\Lambda=\mu(T-t_0)\), the probability of \(t_i\) given \(y_i\) mutations simplifies to: \[ P(t_i | Y_i) = \frac{\mu \Lambda^{y_i}\exp(-\Lambda)}{\gamma(Y_i+1, \Lambda)} \]

• Since we can’t solve for the inverse CDF, we use a simple rejection method to sample integration times.

  • \(t_0\), \(y_i\) and \(\mu\) are sampled from the posterior distribution.

Simulation model

• We used an exact stochastic simulation module in R (twt) to simulate cell population dynamics (forward time).
  • Virus replication completely blocked by ART initiation at time \(t=10\).

Simulating population dynamics

Branching events require a “source” cell to induce a “target” cell to change state. Transmission of virus from an infected cell to a susceptible cell. Transition events occur when a cell switches from one state (compartment) to another. e.g., reactivation of a latently-infected cell. Discrete events determine population dynamics over time.

Simulating trees

From the population size trajectories, twt samples trees in reverse time for given sampling times and cell types. ▲ = branching event (infection from active cell, or clonal expansion of latent cell) ○ = state transition (from active to latent or vice versa) Retaining these tree annotations lets us collapse branches associated with latently-infected cells.

Simulating evolution

• Ran trees from twt through INDELible
  • TN93 nucleotide substitution model with 40% As.
  • Seeded with an HIV-1 subtype C sequence at the root (AY772699)
• Reconstructed trees from simulated alignments with FastTree2 (double precision build).

Comparing bayroot to root-to-tip regression